ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 43, Number 3, 2013 ON THE SOLUTIONS OF DIFFERENCE EQUATIONS OF ORDER FOUR H. EL-METWALLY, E.M. ELSAYED AND E.M. ELABBASY ABSTRACT. In this paper we deal with the behavior of the solution of the following difference equation x n+1 = ax n-1 + bx 2 n-1 cx n-1 + dx n-3 , n =0, 1,..., where the initial conditions x -3 ,x -2 ,x -1 ,x 0 are arbitrary positive real numbers and a, b, c, d are positive constants. Also, we obtain the solution of some special cases of this equation. 1. Introduction. In this paper we deal with the behavior of the solutions of the difference equation (1) x n+1 = ax n-1 + bx 2 n-1 cx n-1 + dx n-3 , n =0, 1,..., where the initial conditions x -3 ,x -2 ,x -1 ,x 0 are arbitrary positive real numbers and a, b, c, d are positive constants. Also, we obtain the solution of some special cases of this equation. Nonlinear rational difference equations are of great importance in their own right because diverse nonlinear phenomena occurring in sci- ence and engineering can be modeled by such equations. Furthermore, the results about such equations offer prototypes towards the develop- ment of the basic theory of nonlinear difference equations. The long term behavior of the solutions of nonlinear difference equa- tions of order greater than one has been extensively studied during the last decade. For example, various results about boundedness, stability and periodic character of the solutions of the second-order nonlinear difference equation see [1 9, 11, 12]. 2010 AMS Mathematics subject classification. Primary 39A10. Keywords and phrases. Periodicity, boundedness, solution of difference equations. Received by the editors on November 9, 2010. DOI:10.1216/RMJ-2013-43-3-877 Copyright c 2013 Rocky Mountain Mathematics Consortium 877